For large-scale applications, there is growing interest in replacing Graph Neural Networks (GNNs) with lightweight Multi-Layer Perceptrons (MLPs) via knowledge distillation. However, distilling GNNs for self-supervised graph representation learning into MLPs is more challenging. This is because the performance of self-supervised learning is more related to the model's inductive bias than supervised learning. This motivates us to design a new distillation method to bridge a huge capacity gap between GNNs and MLPs in self-supervised graph representation learning. In this paper, we propose \textbf{D}iffusion-\textbf{A}ssisted \textbf{D}istillation for \textbf{S}elf-supervised \textbf{G}raph representation learning with \textbf{M}LPs (DAD-SGM). The proposed method employs a denoising diffusion model as a teacher assistant to better distill the knowledge from the teacher GNN into the student MLP. This approach enhances the generalizability and robustness of MLPs in self-supervised graph representation learning. Extensive experiments demonstrate that DAD-SGM effectively distills the knowledge of self-supervised GNNs compared to state-of-the-art GNN-to-MLP distillation methods. Our implementation is available at https://github.com/SeongJinAhn/DAD-SGM.